"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"

Does anyone know where I might find a scanned copy of this or closely related material, in Ramanujan's handwriting? Bruce Berndt kindly pointed me to a typeset version, and much else related of extreme interest -- but I would love to use the original in my lecture, if possible.

Bruce Berndt is probably the mathematician best connected to Ramanujan's legacy, but you could also try the writer David Leavitt via the University of Florida (dleavitt@ufl.edu). He did a lot of digging into archives while writing his novel The Indian Clerk based on the interaction of Hardy and Ramanujan at Cambridge.
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Jim HumphreysDec 2 '11 at 15:43

@Jim: Thank you, that is an excellent idea! I read his book and I was a big fan of it.
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Frank ThorneDec 2 '11 at 16:08

A Disappearing Number is a play which has a scene near the beginning that uses much the same wording as the start of your post. Some undergraduates might find it mathematically inspiring. Gerhard "Ask Me About System Design" Paseman, 2011.12.02
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Gerhard PasemanDec 2 '11 at 16:30

``Srinivasa Ramanujan The Lost Notebook and Other Unpublished Papers," Narosa Publishing House, New Delhi, 1988 has photocopies of several letters from Ramanujan to Hardy, but unfortunately not the one you want.
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Jeff HarveyDec 2 '11 at 18:42

The Wikipedia link seems to be broken. (Not that it's so important, but I had not previously heard that 1+2+3+4+...=-1/12, so I got curious.)
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Hugh ThomasDec 3 '11 at 23:17